Group Analysis and New Explicit Solutions of Simplified Modified Kawahara Equation with Variable Coefficients

Table 9

Optimal system of subalgebras of some special cases of (1.1).

Equation number	Equations	Optimal system
(1.2)	𝑢𝑡+𝑢^2𝑢𝑥+𝑏0𝑢𝑥𝑥𝑥𝑥𝑥=0	𝑉1,𝑉2+𝜆𝑉1,𝑉3
(2.3)	𝑢𝑡+𝑢^2𝑢𝑥+𝑎𝑢+(𝑏0𝑒^−2𝑎𝑡)𝑢𝑥𝑥𝑥𝑥𝑥=0	𝑉1,𝑉2+𝜇𝑉1,𝑉3
(2.4)	𝑢𝑡+𝑢^2𝑢𝑥+𝑎𝑢+(𝑏0𝑒^𝑚𝑡)𝑢𝑥𝑥𝑥𝑥𝑥=0	𝑉1,𝑉2
(3.3)	𝑢𝑡+𝑢^2𝑢𝑥+(1/𝑡)𝑢+(𝑏0/𝑡^2)𝑢𝑥𝑥𝑥𝑥𝑥=0	𝑉1,𝑉2+𝜇𝑉1,𝑉3

Here 𝜇=0,𝜇=±1, and 𝜆 is a nonzero constant.